The Enterprise-A is actually the original Enterprise.
The Enterprise was set to be refit, but Starfleet also started work on the first new Enterprise subclass ship. As both neared the end, it became clear that the Enterprise refit wouldn't be ready to launch the class, so the two were flipped. The new build would become the Enterprise NCC-1701, while the Enterprise herself would be renamed to whatever name that new ship would've been.
However, the refit Enterprise would become a testbed for new technologies over the intervening years. Never actually having an official name though probably having an NX designation.
Then, when Starfleet gets caught over a barrel, not really having a ship to give Kirk and his band of rebels, they pulled the actual refit 1701 off of the testing grounds and gave it to Kirk, to become the 1701-A. Explaining why it was a mess. It was a testbed with tons of conflicting technologies that were never intended to work together over the long term.
So when the seemingly young Enterprise-A is retired at the end of the Battle at Khitomer, it is because it isn't young at all...